Locally Finsler $A$-modules over locally $C*$-algebras

E. Ansari Piri, R. G. Sanati

Suppose $A$ is a Frechet locally $C^*$-algebra and $B$ is a commutative locally $C^*$-algebra. In this paper, we study the notion of locally Finsler module and we show that if $E$ is both a full locally Finsler module over $A$ and $B$ such that there is a map $\varphi: A\rightarrow B$ with the closed range in such a way that $ax=\varphi (a)x$ and $\varphi(\rho A(x))= \rho B(x)$, then $\varphi$ is a continuous and bijective $*$-homomorphism. Moreover, we show that if $\varphi(A)$ is of the second category in $B$, or the locally Finsler seminorms on $E$ are symmetric, then $\varphi$ is $*$-isomorphism of locally $C^*$-algebras. MOSLEHIAN